We study two aspects of discrete-time birth-death processes, the common feature of which is the central role played by the decay parameter of the process. First, conditions for geometric ergodicity and bounds for the decay parameter are obtained. Then the existence and structure of quasi-stationary distributions are discussed. The analyses are based on the spectral representation for the «-step transition probabilities of a birth-death process developed by Karlin and McGregor.

No abstract

Random walk polynomials and random walk measures play a prominent role in the analysis of a class of Markov chains called random walks. Without any reference to random walks, however, a random walk polynomial sequence can be defined (and will be defined in this paper) as a polynomial sequence {P,(x)) which is orthogonal with respect to a measure on [-1, l] and which is such that the parameters a, in the recurrence relations P ,,+ ,(x> = (x-(Y,)P,(x>-/3,P,, _ ,(x) are nonnegative. Any measure with respect to which a random walk polynomial sequence is orthogonal is a random walk measure. We collect some properties of random walk measures and polynomials, and use these findings to obtain a limit theorem for random walk measures which is of interest in the study of random walks. We conclude with a conjecture on random walk measures involving Christoffel functions.